Ask Mr. Smarty Plants

Okay, first, let's make sure what plant we're talking about, as you say you are not sure if you have an euonymus. This Clemson University Extension website has an excellent description of euonymus and recommendations for its care and control of pests. On this site, mention is made of one variety called Euonymus kiautschovicus, which apparently attracts bees and flies during its blooming season in late summer. Take a look at this page of Images of Euonymus kiautschovicus and see if you recognize it as your plant. That was the only mention we found anywhere of flies being attracted to any euonymus plant. According to the USDA Plant Profile for that plant, it is found only in about 5 states, all to the east of you.

Since all of the euonymus species are natives of the Far East, although distributed widely throughout Europe and North America, they are not in our range of expertise at the Lady Bird Johnson Wildflower Center, which is focused on plants native to North America. However, we do try to help gardeners with plants having problems, even when they are not native. Have you also noticed more bees around your plant when it is blooming? Since the blooms are not particularly spectacular, and if you believe that is what is attracting the flies, you might try nipping the blooms in the bud, as it were. That would be a disappointment to the bees who are also coming to your bushes, but at least they wouldn't be killed with an insecticide spray if you went that route. Pruning to eliminate the buds before they bloom would be the least expensive and least damaging to the environment, and is what we would recommend in this case. Try it for a summer, and see if it works.